Hello,

I'm having difficulty in understanding where the book has got these angles from:

Question: Find the maximum and minimum values for $\displaystyle y = 3\sin\theta - 4\cos\theta\ in\ the\ range\ \theta\ to\ 2\pi $

$\displaystyle \frac{d}{d\theta} (3\sin\theta - 4\cos\theta) = 4\sin\theta + 3\cos\theta $

$\displaystyle 4\sin\theta + 3\cos\theta = R\sin(\theta + \alpha) $

$\displaystyle 4\sin\theta + 3\cos\theta = R\sin\theta\cos\alpha + R\cos\theta\sin\alpha $

$\displaystyle R = \sqrt {4^2 + 3^2} = \pm5 $

$\displaystyle \alpha = \arctan \frac{3}{4} \approx 36.869^{\circ} $

$\displaystyle \frac {dy}{d\theta} = 5\sin(\theta + 36.869^{\circ}) = \pm 5\ for\ a\ maximum\ or\ minimum\ value $

$\displaystyle \theta \approx 90 - 36.869^{\circ} \approx 53.131^{\circ}$

Now, in my book the answer is 5 at $\displaystyle 143^{\circ}8' $ for a maximum value, and -5 at $\displaystyle 323^{\circ}8' $ for a minimum value. However, they have subtracted $\displaystyle \alpha $ from $\displaystyle 180^{\circ} $ in order to obtain these pair of angles? I've drawn the function, and these values is where $\displaystyle 5\sin(\theta + 36.869^{\circ}) = 0 $. As a matter of fact, the maximum value first occurs at $\displaystyle 53.131^{\circ} $ and the minimum value at $\displaystyle 233.131^{\circ}$.

I've looked into this a bit deeper, and the only way a maximum or minimum value can occur at these computed angles - is if the first derivative is a negative cosine function.

What have I done wrong?

Thank you for your attention.